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I thought it sort of looked like the Webley Strap as well but it seemed to hang down pretty far.
The box that was referred to is just under his right elbow. It is behind the Pouch and in front of the Thruster. The strap runs right down the center of it in the picture.

Nope, the object on the right side is above the side arm and is all white. Look in the spot that the arrow is pointing to the thruster for the jet pack. It is just to the right of that thruster in the pic. It is white and way too high for the sidearm which is dark in color not white.
In another thread, the discussion about the wire or cord on the left side of Boba that is shown in those Pre-Pro photos has been slightly determined that it is a wire running from the chest lights to a pouch on Boba's hip where the battery is stored. Back when the film was released, they didn't have access to the cool tiny electronics we do today.

First the "white box," is the bottom portion of the plain white or silver shoulder braid. It is the item I have outlined in orange in the photo below

In regard to the additional "cord," yes, that is too long to be the strap for the rifle. However, I don't think it looks like a cord. Looks like a loose strap. I couldn't find any other pics with that strap dangling like that.

Yea,
I am pretty sure Tylerdurden was able to bring us all to light on the pic.
The final determination was that the white box was the bottom tail of the white wookie braid. The odd cord looking thing would be the blaster sling.
The pics you are referring to Phantom are on the Pre-Pro Fett and I did see in another thread a while back that the wire was almost definately the wire leading to the chest light.

i know for a fact that the wire on the left side of the suit IS the wire that runs down to a battery pack / on / off switch and up to the chest display . at the last con i was at , jeremy bulloch took an interest in mine and i told him that i had the wire running down to my side pouch , and he said " that is how we had it set up " . so there you have it , strait from the hunter's mouth .